3 problems

Let $C_N$ denote the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$ of side length $N$, where $d\geq 3$, and let $g(0)$ be the Green's function at the origin for sim…

Let $G_N=(\mathbb{Z}/N\mathbb{Z})^d$ be the discrete torus of dimension $d\geq 3$, let $C_N$ denote the cover time of $G_N$, and let $g(0)$ be the value at the origin of the Green'…

Let $C_3^n$ be the $n$-dimensional discrete torus, and let $G$ be a subgraph of $C_3^n$ with minimum degree at least $d$. The discrete-torus path-length conjecture. $G$ must contai…